Stability Analysis And Controller Design For Fractional Order Systems

Fractional order system is a model system based on the theories of fractional order calculus and fractional differential equation.Differential and integral of general integral is generalized to non-integer,whose order-number is more arbitrary.Thereby the fractional order system possesses stonger desciptivity than integer order system.In another word,fractional order system is one of the particular samples of integer order system.Integer order calculus is decided by the local characteristics of function,its descriptivity is mostly approximate.On the countrary,the fractional order calculus is considered to cover the whole information of function by the weighted method so that it can describe the dynamic response of the actual system more accurately.It requires more strongly to construct the actual system model with the development of the industry.Additionally,integer order calculus can not be applied to construct models for some natural phenomena.Under this background,it becomes quite academically and practically important to study the stability and the control of fractional order system.The following outlined the research:1)The stability of fractional order singular systems was studied.Firstly,based on the state space model of the integer-order generalized system,the theory of the fractional order singular systems was proposed,such as regular,non impulsive and stable.Secondly,by factorizing the coefficient matrix of fractional order generalized system with the method of Weierstrass factorization,and by analysing its solution's uniqueness and impulse response,the necessary and sufficient conditions for the regularity and non-pulse of fractional singular system was proposed.Finally,a sufficient condition for the stability of the fractional order generalized system was obtained by using the D stability theory and the Kronecker product of the matrix.2)The design of the controller for the general fractional order system was researched.Firstly,by using the transfer matrix of fractional order system,the bounded real lemma was brought forward.Then,by judging the characteristic value of the Hamiltion(Hamilton)matrix in the complex plane,the sufficient condition of the fractional order system was worked out based on the LMI form.Finally,the controller was designed for the fractional order system with external disturbance in order to make theclosed-loop system present the H?performance,and to test the effectiveness of the controller design by numerical examples.3)The stability and controller design of the fractional interval uncertain systems were examined.Firstly,based on that the stability region of the order between 1 and 2 of the fractional order linear systems is a convex set,the necessary and sufficient condition for the system stability of the fractional order interval uncertain systems was presented.Then,the design method of output feedback controller was advanced by using the stability condition of the fractional order interval uncertain system.Finally,the feasibility and effectiveness were tested by numerical simulation.